Subgroups — #13506368

Order 1792.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89455	C₂	1	4	yes	maximal
2	#11730752	C₂	1	2	yes	maximal
3	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	#11727520	C₂	1	4	yes	maximal
5	#11727519	C₂	1	2	yes	maximal
6	#11727518	C₂	1	2	yes	maximal
7	C₂×#6433	C₂²	1	4	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
12	C₄³×C₇	C₂²	1	1	yes	characteristic
13	C₂×#6392	C₂²	1	8	yes	—
14	C₂×#6391	C₂²	1	8	yes	—
15	#89708	C₂²	1	4	yes	—
16	#89707	C₂²	1	4	yes	—
17	#89700	—	2	8	no	—
18	#89455	—	2	16	no	—
19	#33311	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	C₂³	1	2	yes	—
21	C₂²×C₇×C₈	C₂³	1	2	yes	—
22	C₂×C₄²×C₇	D₄	1	4	yes	—
23	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₇	—	2	8	no	—
27	C₂×C₄⋊Dic₇	C₂³	1	8	yes	—
28	C₂×Dic₇⋊C₄	—	2	8	no	—
29	#6433	—	2	16	no	—
30	C₄×C₇×C₈	D₄	1	8	yes	—
31	C₄×C₇×C₈	Q₈	1	4	yes	—
32	C₄×C₇×C₈	Q₈	1	4	yes	—
33	C₄×C₇×C₈	Q₈	1	8	yes	—
34	C₇⋊#95	—	2	8	no	—
35	#6392	—	2	16	no	—
36	#6391	—	2	16	no	—
37	C₂×#499	—	7	4	no	—
38	#2859	—	7	2	no	—
39	#1846	—	7	4	no	—
40	#1845	—	7	2	no	—
41	#1844	—	7	2	no	—
42	C₄²×C₈	D₇	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
47	C₂²×Dic₇	—	2	8	no	—
48	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
49	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
50	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
51	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
52	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
53	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
54	C₄²×C₇	—	2	8	no	—
55	C₄²×C₇	C₂×D₄	1	4	yes	—
56	C₄²×C₇	C₂×Q₈	1	4	yes	—
57	C₄²×C₇	C₂×Q₈	1	2	yes	—
58	C₄²×C₇	C₂×Q₈	1	2	yes	—
59	C₄⋊Dic₇	—	2	16	no	—
60	C₄⋊Dic₇	—	4	16	no	—
61	Dic₇⋊C₄	—	4	16	no	—
62	C₂×#125	—	7	4	no	—
63	#499	—	14	16	no	—
64	C₂×#107	—	7	8	no	—
65	C₂×#106	—	7	8	no	—
66	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
69	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
70	#389	—	14	8	no	—
71	#384	—	7	4	no	—
72	#382	—	7	4	no	—
73	C₄³	C₂×D₇	1	1	yes	characteristic
74	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇	—	2	4	no	—
76	C₂×C₄×C₇	—	2	4	no	—
77	C₂×C₄×C₇	#127	1	4	yes	—
78	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
79	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
80	C₂×C₄×C₇	#135	1	8	yes	—
81	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
82	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
85	C₂×Dic₇	—	4	8	no	—
86	C₂×Dic₇	—	4	16	no	—
87	C₇×C₈	C₄⋊Q₈	1	8	yes	—
88	C₇×C₈	C₄⋊Q₈	1	8	yes	—
89	C₂²×C₈	C₂²×D₇	1	2	yes	—
90	C₂²×C₈	C₂²×D₇	1	2	yes	—
91	#125	—	14	16	no	—
92	C₂×C₄⋊C₄	—	7	8	no	—
93	C₂×C₄⋊C₄	—	14	8	no	—
94	C₂×C₄⋊C₄	—	14	8	no	—
95	C₂×C₄²	D₂₈	1	4	yes	—
96	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
99	#107	—	14	16	no	—
100	#106	—	14	16	no	—
101	C₄×C₈	Dic₁₄	1	4	yes	—
102	C₄×C₈	Dic₁₄	1	4	yes	—
103	C₄×C₈	Dic₁₄	1	8	yes	—
104	C₄×C₈	D₂₈	1	8	yes	—
105	#95	—	14	8	no	—
106	C₂²×C₇	C₂×#127	1	1	yes	characteristic
107	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₇	C₂×#135	1	4	yes	—
110	C₄×C₇	—	2	4	no	—
111	C₄×C₇	—	2	4	no	—
112	C₄×C₇	#499	1	8	yes	complemented
113	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
114	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
115	Dic₇	—	8	8	no	—
116	C₂²×C₄	—	14	8	no	—
117	C₂²×C₄	C₂×D₂₈	1	2	yes	—
118	C₂²×C₄	C₂×D₂₈	1	2	yes	—
119	C₂²×C₄	C₂×D₂₈	1	2	yes	—
120	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
122	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
123	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
124	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
125	C₂×C₈	C₂×D₂₈	1	4	yes	—
126	C₂×C₈	C₂×D₂₈	1	4	yes	—
127	C₄⋊C₄	—	14	16	no	—
128	C₄⋊C₄	—	28	16	no	—
129	C₄⋊C₄	—	28	16	no	—
130	C₄²	—	2	8	no	—
131	C₄²	C₂×Dic₁₄	1	4	yes	—
132	C₄²	C₂×Dic₁₄	1	2	yes	—
133	C₄²	C₂×Dic₁₄	1	2	yes	—
134	C₄²	C₂×D₂₈	1	4	yes	—
135	C₂×C₇	#2868	1	1	yes	characteristic
136	C₂×C₇	#3147	1	2	yes	—
137	C₂×C₇	C₂×#499	1	2	yes	—
138	C₂×C₇	#3160	1	2	yes	—
139	C₂³	C₂²×D₂₈	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	28	8	no	—
143	C₂×C₄	—	28	16	no	—
144	C₂×C₄	#6432	1	4	yes	—
145	C₂×C₄	#6432	1	4	yes	—
146	C₂×C₄	#6437	1	4	yes	—
147	C₂×C₄	#6467	1	8	yes	—
148	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₁₄	1	2	yes	—
151	C₂×C₄	C₂²×D₂₈	1	2	yes	—
152	C₈	#6432	1	8	yes	—
153	C₈	#6432	1	8	yes	—
154	C₇	#33311	1	1	yes	characteristic, complemented, Sylow
155	C₂²	C₂×#6432	1	1	yes	characteristic
156	C₂²	C₂×#6432	1	1	yes	characteristic
157	C₂²	C₂×#6437	1	1	yes	characteristic
158	C₂²	C₂×#6467	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	56	8	no	—
162	C₄	#89455	1	8	yes	complemented
163	C₄	C₂×#6432	1	2	yes	—
164	C₄	C₂×#6432	1	2	yes	—
165	C₂	#11730756	1	1	yes	characteristic
166	C₂	C₂×#89455	1	2	yes	—
167	C₂	#11735887	1	2	yes	—
168	C₂	#11735912	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₈
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₄²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	C₄²×C₈	D₇
hypocenter	C₇	#33311
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
112. C₄×C₇	#499	semidirect
154. C₇	#33311	semidirect
162. C₄	#89455	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#89455	2
#11730752	2
C₄²×C₇×C₈	2
#11727520	2
#11727519	2
#11727518	2
#33311	7